Discrepancy of general symplectic lattices

TL;DR

This paper studies the discrepancy in lattice point counts for random symplectic lattices, showing that for almost all such lattices, the discrepancy diminishes polynomially with the volume, extending previous results to a symplectic setting.

Contribution

It extends discrepancy bounds from general lattices to the specific case of symplectic lattices, providing new probabilistic estimates.

Findings

Discrepancy $D(\Lambda,B_t)$ is $O(vol(B_t)^{- ext{delta}})$ for almost every symplectic lattice.

Results generalize Schmidt's bounds from all lattices to symplectic lattices.

Provides probabilistic estimates for lattice point discrepancy in symplectic ensembles.

Abstract

We consider random lattices taken from the general symplectic ensemble and count the number of lattice points of a typical lattice in nested families $B_t$ of certain Borel sets. Our main result is that for almost every general symplectic lattice, the discrepancy $D(\Lambda,B_t)$ of the lattice point count with respect to the volumes is $O(vol(B_t)^{-\delta})$. This extends work of W. Schmidt who gave similar discrepancy bounds for the space of all lattices in $\mathbb{R}^n$.